Sets of constant distance from a compact set in 2-manifolds with a geodesic metric

Abstract:

Let $(M, d)$ be a complete topological 2-manifold, possibly with boundary, with a geodesic metric $d$. Let $X\subset M$ be a compact set. We show then that for all but countably many $\varepsilon$ each component of the set $S(X, \varepsilon )$ of points $\varepsilon$-distant from $X$ is either a point, a simple closed curve disjoint from $\partial M$ or an arc $A$ such that $A\cap \partial M$ consists of both endpoints of $A$ and that arcs and simple closed curves are dense in $S(X, \varepsilon )$. In particular, if the boundary $\partial M$ of $M$ is empty, then each component of the set $S(X, \varepsilon )$ is either a point or a simple closed curve and the simple closed curves are dense in $S(X, \varepsilon )$.